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In this article, we present extensional mappings, that are based on second-order tuple generating dependencies between models in our Generic Role-based Metamodel GeRoMe. Our mappings support data translation between heterogeneous models, such as XML schemas, relational schemas, or OWL ontologies. The mapping language provides grouping functionalities that allow for complete restructuring of data, which is necessary for handling object oriented models and nested data structures such as XML. Furthermore, we present algorithms for mapping composition and optimization of the composition result. To verify the genericness, correctness, and composability of our approach we implemented a data translation tool and mapping export for several data manipulation languages. Furthermore, we address the question how generic schema mappings can be harnessed for answering queries against an integrated global schema.